Gears and Gear Trains

Chapter 13

Part 1

Types of Gears

Types of Gears

Spur Helical

Bevel Worm

Nomenclature of Spur-Gear Teeth

Shigley’s Mechanical Engineering DesignFig. 13–5

Rack

A rack is a spur gear with an pitch diameter of infinity.

The sides of the teeth are straight lines making an angle to the line

of centers equal to the pressure angle.

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Fig. 13–13

Tooth Size, Diameter, Number of Teeth

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Tooth Sizes in General Industrial Use

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Table 13–2

Modern Gear Teeth Have an Involute (Tooth) Profile

The most common conjugate action tooth profile

is the involute profile.

Can be generated by unwrapping a string from a cylinder, keeping

the string taut and tangent to the cylinder.

Circle is called base circle.

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How an Involute Gear Profile is constructed

If you tied a string around a base circle, then keep the string pulled tight.

Move the string off the circle, starting at point A0 and sketch out the profile

from that point on the string: A0 to B1 to B2 to B3 to B4

A1B1=A1A0, A2B2=2 A1A0 , etc

Why would we want an involute tooth surface?

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• Constant speed ratio between gears

• Smoother speeds

• Constant line of action of forces between gears

• Which means constant torque transmitted

• Theoretical gear-to-gear contact at tangents of

pitch circles

• Allow multiple teeth to be in contact

• Higher loads (spreads loads over more teeth)

• Tolerant of center distance errors, keeps constant

speed even if shaft/gears are not exactly perfect.

Spur Gear forces are along the Pressure line

Pressure Angle Φ has the values of 20° or 25 °

14.5 ° has also been used.

Pressure Angle remains constant, a function of the involute

profile.

Relation of Base Circle to Pressure Angle

& to the Pitch Circle

Shigley’s Mechanical Engineering DesignFig. 13–10

Standardized Tooth Systems: AGMA Standard

Common pressure angles f : 20º and 25º

Older pressure angle: 14 ½º

Common face width, F=width:

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3 5

3 5

p F p

pP

FP P

=

Gear Sources

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• Boston Gear

• Martin Sprocket

• W. M. Berg

• Stock Drive Products

….

Numerous others

Conjugate Action

When surfaces roll/slide

against each other and

produce constant angular

velocity ratio, they are said

to have conjugate action.

Can be accomplished if

instant center of velocity

between the two bodies

remains stationary between

the grounded instant centers.

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Fig. 13–6

Fundamental Law of Gearing:

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The common normal of the tooth

profiles at all points within the mesh

must always pass through a fixed point

on the line of the centers called pitch

point. Then the gearset’s velocity ratio

will be constant through the mesh and

be equal to the ratio of the gear radii.

Conjugate Action: Fundamental Law of Gearing

Forces are transmitted on line of action which is normal to the contacting surfaces.

Velocity. VP of both gears is the same at point P, the pitch point

Angular velocity ratio is inversely proportional to the radii to point P, the pitch point.

Circles drawn through P from each fixed pivot are pitch circles, each with a pitch radius.

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Fig. 13–6

VP

Gear Ratio

𝑉𝑃 = 𝜔1𝑟1=𝜔2𝑟2

𝜔1

𝜔2=

𝑟2

𝑟1=

𝑁2

𝑁1

Gear Ratio >1

VP of both gears is the same at point P, the pitch (circle contact) point

ω1

ω2

r2

r1

P

N2N1

ω2 rotates opposite of ω1

Pitch Circle of Gears

Nomenclature

Smaller Gear is Pinion and Larger one is the gear

In most application the pinion is the driver, This reduces speed

but it increases torque.

Simple Gear Trains

For a pinion 2 driving a gear 3, the speed of the driven gear is

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n2 =ω2

n3 =ω3

r3

r2

P

N3N2

VP

Simple Gear Trains (Gear 2 to Gear 4)

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Fig. 13–27

Compound Gear Train

A practical limit on train value for one pair of gears is 10 to 1

To obtain more, compound two gears onto the same shaft

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Fig. 13–28

Compound Gear Trains

))()((5

4

4

3

2

1

1

5

N

N

N

N

N

N

n

n−−−=

Example 13–3

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Example 13–4

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Example 13–4

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Example 13–5

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Example 13–5

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Example 13–5

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Example 13–5

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Example 13–5

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